# Is Modulo Multiplication and Addition Associative, Distributive, and Commutative?

Created by Anna Szczepanek, PhD
Reviewed by Rijk de Wet
Last updated: Feb 01, 2022

Let's discuss the algebraic properties of the integer modulo addition and multiplication operations — their associativity, distributivity, and commutativity. We will also briefly explain what each of these properties means in algebra.

## Addition and multiplication modulo n

Select a non-zero integer n. The symbol [x] will denote the set of all integers congruent to x mod n, i.e. the numbers of the form x + n*y, where y is an integer.

• The "addition modulo n" operation is defined as [a]+[b] = [a+b]. In other words:

(a + b) mod n = (a mod n + b mod n) mod n.

• The "multiplication modulo n" operation is defined as [a]*[b] = [a*b]. So:

(a * b) mod n = ((a mod n) * (b mod n)) mod n.

We will now discuss various properties of both these modular operations.

## Is modular arithmetic associative?

Associativity means that the result will not change when we rearrange the parentheses in an expression. It turns out that:

([x] + [y]) + [z] = [x] + ([y] + [z])

• Modular multiplication is also associative:

([x] * [y]) * [z] = [x] * ([y] * [z])

In the next section, we prove that modular multiplication is associative.

## Proof that multiplication modulo n is associative

We will now prove that

([x] * [y]) * [z] = [x] * ([y] * [z]).

Let's start from the left-hand side. Below, each line is equivalent to the preceding one.

([x] * [y]) * [z]

By the definition of modular multiplication, we get:

([x * y]) * [z]

Again by the definition of modular multiplication:

[(x * y) * z]

We use the associativity of the multiplication of real numbers:

[x * (y * z)]

By the definition of modular multiplication again, we get:

[x] * ([y * z])

Again by the same definition:

[x] * ([y] * [z])

And look, we have arrived at the right-hand side. Hence, we have proved that multiplication modulo n is associative!

## Is modulo arithmetic commutative?

Commutativity means that the result will not change when we change the order of the operands. One can easily show that:

([x] + [y]) + [z] = [x] + ([y] + [z])

• Modular multiplication is also associative:

([x] * [y]) * [z] = [x] * ([y] * [z])

The proof is very similar to what we've seen above for associativity. This time, you'll need to use the fact that the multiplication/addition of real numbers is commutative.

## Is modulus function distributive?

Distributivity is a property that involves both addition and multiplication at once. We say that multiplication distributes over addition if instead of multiplying a sum of several terms by a factor, we can multiply each summand by this factor individually and then add these partial results together to obtain the final answer. So, for example, 5*12 = 5*(10+2) = 5*10 + 5*2 = 60.

It turns out that modular multiplication is distributive over addition:

([x] + [y]) * [z] = [x] * [z] + [y] * [z]

and

[x] * ([y] * [z]) = [x] * [y] + [x] * [y]

In proving this you'll need to evoke the fact that for real numbers multiplication distributes over addition.

Anna Szczepanek, PhD
x mod y = r
x (dividend)
y (divisor)
r (remainder)
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